Dispersive estimates and generalized Boussinesq equation on hyperbolic spaces with rough initial data

Abstract

We consider the generalized Boussinesq (GBq) equation on the real hyperbolic space $\mathbb{H}^{n}$ ($n\geq2$) in a rough framework based on Lorentz spaces. First, we establish dispersive estimates for the GBq-prototype group, which is associated with a core term of the linear part of the GBq equation, through a manifold-intrinsic Fourier analysis and estimates for oscillatory integrals in $\mathbb{H}^{n}$. Then, we obtain dispersive estimates for the GBq-prototype and Boussinesq groups on Lorentz spaces in the context of $\mathbb{H}^{n}$. Employing those estimates, we obtain local and global well-posedness results and scattering properties in such framework. Moreover, we prove the polynomial stability of mild solutions and leverage this to improve the scattering decay.

Theorems & Definitions (17)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Remark 3.4
• Lemma 3.5
• Theorem 4.1: Local-in-time well-posedness
• Remark 4.2